Breakup in nucleon-deuteron scattering with Δ-isobar excitation

Breakup in nucleon-deuteron scattering with ⌬ -isobar excitation

K. Chmielewski,^1,*A. Deltuva,^1,†A. C. Fonseca,²S. Nemoto,³ and P. U. Sauer¹

1Institut fu¨r Theoretische Physik, Universita¨t Hannover, D-30167 Hannover, Germany

2Centro de Fı´sica Nuclear da Universidade de Lisboa, P-1649-003 Lisboa, Portugal

3Department of Physics, Faculty of Science and Technology, Science University of Tokyo, Noda, Chiba 278, Japan 共Received 8 July 2002; published 22 January 2003兲

Breakup in nucleon-deuteron scattering is described. The description is based on a coupled-channel two- baryon potential that allows for the virtual excitation of a nucleon to a⌬isobar. The Coulomb interaction is not included. Channel coupling gives rise to an effective three-nucleon force. The three-particle scattering equa- tions are solved by real-axis integration using a separable expansion of the two-baryon transition matrix.

Examples for spin-averaged and spin-dependent observables are calculated and compared with experimental data.

DOI: 10.1103/PhysRevC.67.014002 PACS number共s兲: 21.45.⫹v, 21.30.⫺x, 25.10.⫹s, 24.70.⫹s

I. INTRODUCTION

This paper is the third in a series on nucleon-deuteron scattering. The first one 关1兴, called paper I, establishes a separable expansion for the underlying two-baryon interac- tion关2兴, which explicitly allows for⌬-isobar excitation. The second one关3兴, called paper II, uses that separable expansion of the coupled-channel transition matrix for the calculation of elastic nucleon-deuteron scattering, below and above three-nucleon breakup. This paper does so for inelastic nucleon-deuteron scattering, i.e., for three-nucleon breakup.

The ⌬ isobar gives rise to an effective three-nucleon force.

The calculation is without Coulomb interaction. Thus, it re- fers to breakup in neutron-deuteron scattering, though the comparison is mostly with data of proton-deuteron scatter- ing.

The theoretical description of elastic nucleon-deuteron scattering up to about 150 MeV nucleon lab energy in terms of realistic two-nucleon potentials has been generally quite successful关4,5兴with the following exceptions.

共1兲At low energies the description of scattering observ- ables and of bound-state properties are correlated. An appro- priate three-nucleon force has to be added to account for trinucleon binding in full.

共2兲The description of proton-deuteron scattering at very low energies for most angles and at higher energies predomi- nantly in forward direction requires the inclusion of the Cou- lomb interaction between the protons.

共3兲There are long-standing discrepancies in the spin ob- servables A_y(n) and iT₁₁around 10 MeV neutron lab energy.

Furthermore, without a three-nucleon force, the minimum of the unpolarized differential cross section beyond 65 MeV nucleon lab energy cannot be accounted for; this fact is called Sagara discrepancy.

Paper II and Ref. 关6兴study the effect of the⌬ isobar on elastic nucleon-deuteron scattering. The effect is usually small; at most, modest for some spin observables at higher

energies. Reference 关6兴finds a beneficial ⌬-isobar effect on the Sagara discrepancy. The ⌬ isobar is not helpful for the spin observables A_y(n) and iT₁₁around 10 MeV neutron lab energy.

This paper extends the description to spin-averaged and spin-dependent observables of breakup in nucleon-deuteron scattering. Experimental data for breakup are much scarcer than for elastic scattering. Kinematical regimes in which the three-nucleon force mediated by the ⌬ isobar may play a determining role are searched for. The theoretical description requires a change of technique when solving the three- particle scattering equations compared with paper II, which employed a contour deformation technique. Real-axis inte- gration is used instead. The technique is developed in the present context.

In Sec. II basic features of the calculation are described;

however, the important technical details are deferred to the Appendix. Section III presents our results for spin-averaged and spin-dependent observables of breakup in inelastic nucleon-deuteron scattering. The conclusions are given in Sec. IV.

II. BASIC FEATURES OF THE CALCULATION The notation is taken over from paper I and is assumed to be self-evident; explanations of the notation are kept to a minimum.

A. Alt-Grassberger-Sandhas„AGS…breakup equation The symmetrized break-up transition matrix U₀(Z) is de- fined in Eq.共2.13兲of paper I according to

U₀共Z兲⫽G₀^⫺1共Z兲⫹关1⫹T_␣共Z兲G₀共Z兲兴U共Z兲. 共1a兲 It is obtained from the symmetrized multichannel transition matrix U(Z),

U共Z兲⫽PG₀^⫺1共Z兲⫹PT_␣共Z兲G₀共Z兲U共Z兲, 共1b兲 given in Eq. 共2.12兲 of the same paper. Using Eq.共1b兲once more, we rewrite the standard quadrature for the breakup transition matrix as follows:

*Electronic address: Karsten.Chmielewski@itp.uni-hannover.de

†On leave from Institute of Theoretical Physics and Astronomy, Vilnius University, Vilnius 2600, Lithuania.

U₀共Z兲⫽共1⫹P兲G₀^⫺1共Z兲⫹共1⫹P兲T_␣共Z兲G₀共Z兲U共Z兲, 共2兲 where G₀(Z) is the free resolvent, P⫽P₁₂₃⫹P₃₂₁the per- mutation operator, and T_␣(Z) the two-baryon coupled- channel transition matrix between the baryons␤ ^and␥ ^{in the} three-particle space, (␣␤␥) cyclic. The free Hamiltonian in G₀(Z) does not contain center of mass 共c.m.兲 motion, but baryon rest masses, normalized to zero for three nucleons.

The channel states兩␾_␣^(q)␯_␣典^and兩␾0(pq)␯0典 are defined in paper I, p and q are the internal Jacobi momenta,␯␣ and␯0

are sets of discrete quantum numbers determining the chan- nel states in full. Both channel states are antisymmetrized with respect to the pair (␤␥^). 兩␾_␣^(qi)␯_␣i典 is the initial nucleon-deuteron state with the initial c.m. energy E_i⫽e_d

⫹q_i²/2 M_␣, e_d being the deuteron binding energy, M_␣

⫽2m_N/3 the reduced spectator mass, and m_N the nucleon rest mass. 兩␾0(pq)␯0典 is the final three-nucleon breakup state. It is given in paper I as a coupled state with respect to pair spin and isospin. However, the final state is measured in the uncoupled form, i.e.,

兩␾0共p_fq_f兲␯0共m_f兲典^⫽1⫺

冑

^{P2␤␥兩pfqf典␣兩s␣ms␣ft␣mt␣fb␣f典兩s␤ms␤ft␤mt␤fb␤f典兩s␥ms␥ft␥mt␥fb␥f典, 共3a兲}

兩␾0共p_fq_f兲␯0共m_f兲典⫽S_fM

兺

_s

fT_fM_T

f

兩␾0共p_fq_f兲␯0_f典具^s␤m_s

␤_fs_␥m_s

␥_f兩S_fM_S

f典具^t␤m_t

␤_ft_␥m_t

␥_f兩T_fM_T

f典^. 共3b兲

The discrete quantum numbers of the final state are explained in Fig. 2 of paper I. Its c.m. energy is E_f⫽p²_f/2␮_␣⫹q²_f/2 M_␣ with the reduced pair mass ␮_␣⫽m_N/2.

The S matrix for breakup is given by the symmetrized on-shell breakup transition matrix U₀(Z), i.e.,

具␾0共p_fq_f兲␯0共m_f兲兩S兩␾␣共q_i兲␯␣_i典^⫽⫺2␲ⁱ␦共E_f⫺E_i兲具␾0共p_fq_f兲␯0共m_f兲兩U₀共E_i⫹i0兲兩␾␣共q_i兲␯␣_i典^{. 共4兲}

When determining the S matrix the initial and final states are fully antisymmetrized and normalized through (1⫹P)/

冑

^3;

however, those symmetrization operators are incorporated into the definition of the symmetrized breakup transition ma- trix U₀(Z) of paper I. The on-shell transition matrix U₀(Z) is calculated according to Eq.共2兲.

B. Separable expansion of coupled-channel interaction and AGS breakup equation

The form 共2兲 of the breakup transition matrix U₀(Z) is especially convenient, when the two-baryon transition matrix T_␣(Z) is separably expanded according to our general strat- egy for solving the AGS three-particle scattering equations.

The separable expansion

T_␣共Z兲⫽兩g_␣典^⌻␣共Z兲具^g␣兩 共5a兲 yields for U₀(Z),

U₀共Z兲⫽共1⫹P兲G₀^⫺1共Z兲⫹共1⫹P兲兩g_␣典^⌻␣共Z兲

⫻具^g␣兩G₀共Z兲U共Z兲. 共5b兲

Since the deuteron state defines one element, labeled i₀, in the form factor兩g_␣典 of the separable expansion共5a兲,⌻_␣(Z) being the corresponding propagator, the initial nucleon- deuteron state can be rewritten as

兩␾_␣共q_i兲␯_␣i典

⫽G₀共E_i⫹i0兲兩g_␣^(i0␲0I0T0)M_I

iM_T

0典兩q_is₀m_s

it₀m_t

0b₀典␣. 共6兲 Thus, the breakup S matrix共4兲based on the breakup transi- tion matrix U₀(Z) in the form共2兲needs the matrix elements of the operator 具^g␣兩G₀(Z)U(Z)G₀(Z)兩g_␣典 half-shell. Paper II calculated the same operator on shell for the description of elastic nucleon-deuteron scattering.

C. Solution of the integral equation for the half-shell transition matrixŠg_␣円G₀„Z…U„Z…G₀„Z…円g_␣‹

The transition matrix 具^g␣兩G₀(E_i⫹i0)U(E_i⫹i0)G₀(E_i

⫹i0)兩g_␣典is required half-shell for the on-shell breakup am- plitude U₀(E_i⫹i0)兩␾_␣^(qi)␯␣_i典 according to Eqs. 共5b兲 and 共6兲. It is obtained by solving the integral equation

具^g␣兩G₀共Z兲U共Z兲G₀共Z兲兩g_␣典⫽具^g␣兩PG₀共Z兲兩g_␣典

⫹具^g␣兩PG₀共Z兲兩g_␣典^⌻␣共Z兲

⫻具^g␣兩G₀共Z兲U共Z兲G₀共Z兲兩g_␣典^. 共7兲 The kernel 具^g␣兩PG₀(Z)兩g_␣典^⌻␣(Z) of the integral equation 共7兲 contains singularities: 具^g␣兩PG₀(Z)兩g_␣典 develops so- called moving singularities of kinematical origin above the breakup threshold, whereas the propagator ⌻_␣(Z) contains

the deuteron bound-state pole. The arising of these singulari- ties was discussed in depth in paper II which employed the method of contour deformation for dealing with them. That method was adequate for the calculation of on-shell matrix elements needed for the description of elastic scattering in paper II, but it was already tedious there. For breakup calcu- lations that method gets even more tedious. It requires at least two distinct complex paths, and those paths have to be different for different available energies E_i. Contour defor- mation for breakup has not been numerically successful in the past. It was also tried by us tentatively, but problems of stability convinced us to develop the alternative technique of real-axis integration for solving Eq. 共7兲. Its implementation rests on three technical pillars: spline interpolation, numeri- cal evaluation of the singular integrals by specially calcu- lated weights, and Pade´ approximation.

The details of the method are described in the Appendix;

all items have novel aspects. The reliability of the technique is tested by comparing results for elastic nucleon-deuteron scattering which were obtained with the contour-deformation technique in paper II. In fact, all results given there in plots were recalculated using the real-axis technique. No differ- ences, visible in plots, could be found, except minute ones for some spin observables of particularly small magnitude.

Hence, no samples of that comparison are shown in this pa- per. The reliability check is an internal one for elastic nucleon-deuteron scattering; the comparison is possible for the coupled-channel interaction with⌬-isobar excitation em- ployed 关2兴 and for its purely nucleonic reference potential, the Paris potential 关7兴. The method of real-axis integration can without technical change be carried over from elastic nucleon-deuteron scattering to breakup. Furthermore, in the case of the purely nucleonic Paris potential, the comparison is also possible with the breakup results of Ref. 关5兴. This comparison turned out to be quite satisfactory. Thus, we con- sider our technique of real-axis integration highly reliable, and we employ it in this paper for calculating breakup in nucleon-deuteron scattering and for studying ⌬-isobar ef- fects in that process.

D. Observables of nucleon-deuteron breakup

The calculations of this paper are entirely nonrelativistic.

Nevertheless, we like to make the step to observables by starting out from the relativistic form of the cross section,

d␴i→f⫽兩具^f兩M兩i典^{兩2dLips共k␣i⫹kd,k␣f,k␤f,k␥f兲} 4c²

冑

^共k␣i•k_d兲²⫺m_N²m_d²c⁴ . 共8a兲

The reason is that we carry out corresponding calculations of electromagnetic processes; for them the relativistic form of the cross section has important conceptual advantages. In Eq.

共8a兲具^f兩M兩i典 is the Lorentz-invariant singularity-free matrix element, dLips(k_␣

i⫹k_d,k_␣

f,k_␤

f,k_␥

f) the Lorentz-invariant phase space element of the final state defined in Eq.共11兲, and 4c²

冑

^(k␣i•k_d)²⫺m_N²m_d²c⁴ a Lorentz-invariant factor con- taining the initial-state information.

We use the cross section共8a兲in the lab system. The target deuteron is at rest, i.e., k_d⫽0, the impinging nucleon has momentum k_␣

i, which defines the z direction. The changes that arise when the deuteron impinges on a nucleon target are obvious. The matrix element具^f兩M兩i典 ^{of Eq.}共8a兲should be derived from a fully relativistic description of hadron dy- namics. We are unable to give such a relativistic description.

The nonrelativistic hadron dynamics employed is based on a two-baryon potential, fitted to data with the nonrelativistic form of the cross section in contrast to Eq.共8a兲, it connects the S matrix with the symmetrized on-shell breakup transi- tion matrix U₀(E_i⫹i0) in Eq.共4兲; it uses nonrelativistic en- ergies for E_iand E_f. When, nevertheless, that breakup tran- sition matrix is taken for an approximate construction of 具^f兩M兩i典, ignoring the difference in kinetic energies for a relativistic S matrix and its nonrelativistic correspondence of Eq. 共4兲, the following identification is obtained:

具^f兩M兩i典⫽具␾0共p_fq_f兲␯0共m_f兲兩U₀共E_i⫹i0兲兩␾_␣共q_i兲␯_␣i典

⫻共2␲ប兲^9/2 បc

冑

^2k␣i

0c2k_d⁰c2k_␣

f 0 c2k_␤

f 0 c2k_␥

f 0 c.

共8b兲 The calculation of 具␾0(p_fq_f)␯0(m_f)兩U₀(E_i

⫹i0)兩␾_␣^(qi)␯_␣i典 uses the available initial c.m. energy E_i and the Jacobi momenta q_i, p_f, and q_f; their relations to the single particle lab momenta are

E_i⫽e_d⫹ k_␣

i 2

3m_N, 共9a兲

q_i⫽⫺2 3k_␣

i, 共9b兲

p_f⫽1 2共k_␤

f⫺k_␥

f兲, 共9c兲

q_f⫽共k_␤

f⫹k_␥

f兲⫺2

3k_␣

i. 共9d兲

The employed nonrelativistic dynamics is Galilean invariant.

This implies that the matrix element 具^f兩M兩i典 is frame de- pendent. The frame dependence of Eq. 共8b兲 is due to the energy factors

冑

^2k␣i

0 c2k_d⁰c2k_␣

f 0 c2k_␤

f 0 c2k_␥

f

0 c; they arise, rather artificially in Eq. 共8b兲, since corresponding factors have to be attached to the phase space part of the cross sec- tion共8a兲. We note that already at 65 MeV nucleon lab energy the difference between lab and c.m. system amounts to a frame dependence of 2.5% for 具^f兩M兩i典. For the description of spin-averaged and spin-dependent cross sections the breakup transition matrix U₀(E_i⫹i0) is conveniently abbre- viated by the scattering amplitude M (E_ip_fq_f),

具␾0共p_fq_f兲␯0共m_f兲兩U₀共E_i⫹i0兲兩␾␣共q_i兲␯␣_i典

⫽具^ms_␣

f

m_s

␤_fm_s

␥_f兩M共E_ip_fq_f兲兩M_I

im_s

i典^, 共10兲

in which the dependence on the spin projections of the par- ticles in the initial and final states is made explicit. The neu- tron and proton nature of the nucleons (␣␤␥) in the final state is notationally not indicated, but always determined by experiment.

In contrast to the matrix element具^f兩M兩i典 that carries the dynamics, the kinematical factors in Eq. 共8a兲, i.e., the Lorentz-invariant phase-space element

dLips共k_␣

i⫹k_d,k_␣

f,k_␤

f,k_␥

f兲

⫽共2␲ប兲⁴␦⁽⁴⁾共k_␣

f⫹k_␤

f⫹k_␥

f⫺k_␣

i⫺k_d兲

⫻ d³k_␣

fd³k_␤

fd³k_␥

f

共2␲ប兲⁹2k_␣

f 0 c2k_␤

f 0 c2k_␥

f

0 c 共11兲

and the factor 4c²

冑

^(k␣i•k_d)²⫺m_N²m_d²c⁴, which contains the incoming flux, the target density, and projectile and target normalization factors, could, in principle, be calculated rela- tivistically. We shall not use that option in this paper; we believe that it is not justified; we discuss the reason in more detail in Sec. II E.

The momenta in the initial and final states are constrained by energy and momentum conservation. For example, if the momentum k_␤

f and the direction kˆ_␥

f were measured, all three nucleon momenta are determined in the final state, al- though not always uniquely. In practice, the two nucleon scattering angles with respect to the beam direction (␪␤,␸␤) and (␪_␥^,␸_␥), usually notationally shortened to (␪_␤^,␪_␥^,␸_␥

⫺␸␤), and their kinetic energies without rest masses, E_␤

f

and E_␥

f, are meassured. Those energies are related by mo- mentum and energy conservation and therefore lie on a fixed kinematical curve. The observables are therefore given as functions of the arclength S along that curve, i.e.,

S⫽

冕

0 S

dS 共12兲

with dS⫽

冑

^dE␤f 2 ⫹dE_␥

f 2 and E_␥

f being considered a function of E_␤

f or vice versa depending on numerical convenience.

The arclength is always taken counterclockwise along the kinematical curve. No confusion between the arclength S and the S matrix of Eq.共4兲should arise. The normalization of the arclength value zero is chosen differently in different kine- matical situations.

The lab cross section therefore takes the compact form

d␴i→f⫽兩具^ms_␣

f

m_s

␤_fm_s

␥_f兩M共E_ip_fq_f兲兩M_I

im_s

i典兩²

⫻fps dSd²kˆ_␤

fd²kˆ_␥

f 共13a兲

with the abbreviation fps for the phase-space factor. Using relativistic kinematics it takes the following form, i.e.,

fps⫽共2␲兲⁴ប² k_␣

i 0

兩k_␣

i兩c

冕

^{d3k␣fk␥2fdk␥f}

冉

^{k␤2dSfdk␤f}

冊

^{␦共E␣f⫹E␤f}

⫹E_␥

f⫺e_d⫺E_␣

i兲␦共k_␣

f⫹k_␤

f⫹k_␥

f⫺k_␣

i兲. 共13b兲

Here, E_␣

i and E_␣

f are kinetic energies, defined correspond- ingly to E_␤

f and E_␥

f; (2␲ប)^⫺3兩k_␣

i兩c/k_␣

i

0 is the incoming flux in the lab system; the energy factors contained in 具^f兩M兩i典 ^{of Eq.} 共8b兲 and in d Lips(k_␣

i⫹k_d,k_␣

f,k_␤

f,k_␥

f) of

Eq. 共11兲 cancel exactly, once both are assumed to be com- puted in the same frame. The cross section共13a兲is still spin dependent.

The spin-averaged fivefold differential cross section is d⁵␴

¯ dSd²kˆ_␤

fd²kˆ_␥

f

⫽1 6 _M

兺

I_im_s

i

m_s

兺

␣f

m_s

␤f

m_s

␥f

d⁵␴i→f

dSd²kˆ_␤

fd²kˆ_␥

f

⫽1

6Tr关M共E_ip_fq_f兲M^†共E_ip_fq_f兲兴 fps.

共14兲 In the figures the spin-averaged fivefold differential cross section is denoted by d⁵␴^/dSd⍀1d⍀2, the traditional nota- tion.

The spin dependence of the initially prepared states is described by the Hermitian density matrix␳i, normalized to Tr␳i⫽1. The density matrix ␳i of the initially prepared states is the tensor product of density matrices for the nucleon and the deuteron, ␳^{n and}␳^{d, i.e.,}

␳i⫽␳ⁿ丢␳^d. 共15兲 Their individual spin dependence is carried by the spin-¹₂ operators S^a2 and the spin-1 operators S^a3, defined in Sec.

3.2 of paper II. As in paper II, the set of product operators 兵^Sai其^⫽兵^Sa2丢Sa3其 is formed. They are normalized by

Tr关S^aiS^bi兴⫽6␦a_ib_i. 共16兲 With those product operators S^ai the initial density matrix gets the concise form

␳i⫽1 6

兺

_a

i

Tr关␳iS^ai兴S^ai. 共17兲 The final-state polarization measurement is described by the projection operator ␳f, i.e., ␳f

2⫽␳f, which is the tensor product of corresponding projection operators for the three nucleons, i.e.,

␳f⫽N_f␳ⁿ丢␳ⁿ丢␳^n, 共18兲 with N_f⫽2^3⫺N, N being the number of polarization mea- surements.␳fis normalized to Tr␳f⫽N_f. Equation共18兲cor- rects the imprecise description of this point in paper II. The operators ␳ⁿ are parametrized in the form of the nucleon density matrix. Their individual spin dependence is carried

by the spin-¹₂ operators S^a2. The set of product operators 兵^Saf其^⫽兵^{Sa2丢Sa2丢Sa2}其 is formed, which are normalized by

Tr关S^afS^bf兴⫽8␦a_fb_f. 共19兲 With these product operators S^af the projection operator ␳f

gets the concise form

␳f⫽1 8

兺

_a

f

Tr关␳fS^af兴S^af. 共20兲 In terms of the scattering amplitude M (E_ip_fq_f), of the initial density matrix␳i and of the final-state projection op-

erator ␳f, the spin-dependent differential cross section be- comes

d⁵␴ dSd²kˆ_␤

fd²kˆ_␥

f

⫽Tr关M共E_ip_fq_f兲␳iM^†共E_ip_fq_f兲␳f兴 fps.

共21兲 Using the spin-averaged differential cross section d⁵␴

¯ /dSd²kˆ_␤

fd²kˆ_␥

f of Eq.共14兲and the expansions共17兲and 共20兲 for the initial density matrix␳i and the final-state pro- jection operator ␳f, the spin-dependent differential cross sections take the form

d⁵␴ dSd²kˆ_␤

fd²kˆ_␥

f

⫽ ^¯d⁵␴ dSd²kˆ_␤

fd²kˆ_␥

f

1 8 _a

兺

ia_f

Tr关␳iS^ai兴Tr关␳fS^af兴Tr关M共E_ip_fq_f兲S^aiM^†共E_ip_fq_f兲S^af兴

Tr关M共E_ip_fq_f兲M^†共E_ip_fq_f兲兴 . 共22兲

Characteristic for the experimental setup of the studied reaction are the parameters in the initial density matrix ␳i

and in the final-state projection operator␳f which determine the expansion coefficient Tr关␳iS^ai兴Tr关␳fS^af兴 in Eq. 共22兲. Characteristic for the spin dependence of the reaction mecha- nism is the way in which the spin operators S^aiof␳iand S^af of ␳f weigh the spin matrix elements of the scattering am- plitude. The experiment therefore aims at determining ob- servables of the type Tr关M (E_ip_fq_f)S^aiM^†(E_ip_fq_f)S^bf兴/ Tr关M (E_ip_fq_f) M^†(E_ip_fq_f)兴. A particular choice of the spin operators S^ai and S^bf defines particular spin observables;

their notation is standardized in Ref.关8兴.

E. Problem in the comparison of theoretical predictions and experimental data

The experimental setup for breakup usually works with two particle detectors at two fixed angles measuring kˆ_␤

f and

kˆ_␥

f and determines cross sections as functions of the ar- clength S on the kinematical curve corresponding to the two kinetic energies E_␤

f and E_␥

f. A sound comparison requires the same kinematical curve for the experimental interpreta- tion of data and for the theoretical prediction. However, the experimental interpretation of data usually prefers relativistic kinematics, whereas theory prefers nonrelativistic kinemat- ics, since the description of dynamics is nonrelativistic any- how. Without a relativistic treatment of the dynamics there is no fully consistent description of the experimental data and of the theoretical prediction. Thus, approximative identifica- tion procedures have to be applied; a discussion of this point and a suggestion for identification is given in Ref. 关9兴. We follow a somehow different procedure. At the rather low en- ergies considered in this paper the resulting kinematical curves, defined in Eq. 共12兲, are often quite similar for rela-

tivistic and nonrelativistic kinematics, but there are special situations with dramatic differences. Figures 1 and 2 give examples for either case at 65 MeV nucleon lab energy and at 52 MeV deuteron lab energy, respectively.

Figure 1 refers to the space star configuration at 65 MeV nucleon lab energy, which is realized for relativistic and non- relativistic kinematics at slightly different scattering angles.

There are only minor differences between the relativistic and nonrelativistic kinematical curves corresponding to the same angles. However, the kinematical curves for slightly different angles corresponding to the exact space star configuration with relativistic and nonrelativistic kinematics are even al- most identical. The right-hand side of Fig. 1 shows a sample effect on observables, which arises from differences in the kinematical curves. Correspondence is obtained by scaling all considered kinematical curves to the length of the relativ- istic arclength. The length of the kinematical curves before scaling is recorded in the figure caption; the discrepancy be- tween the results of different identification procedures is small.

The example of Fig. 2 is more dramatic. It refers to the quasi-free-scattering 共QFS兲 configuration for 52 MeV deu- teron lab energy. Again, this special situation is with relativ- istic and nonrelativistic kinematics realized only for slightly different scattering angles. However, in this case there are quite large differences between the relativistic and non- relativistic kinematical curves corresponding to the same angles; the reason is that the critical situation (42.26°,42.26°,180.0°), at which the relativistic locus col- lapses to a point, is near and that in nonrelativistic kinemat- ics that critical situation occurs at larger angles. In contrast, the kinematical curves for slightly different angles corre sponding to the exact QFS configuration with relativistic and nonrelativistic kinematics are quite close. The right-hand side of Fig. 1 shows a sample effect on observables, which

arises from differences in the kinematical curves. Correspon- dence is naturally achieved wihout scaling, since the experi- mental data at this energy are given and will be given as functions of S/S_max, S_maxbeing the full arclength of the rela- tivistic kinematical curve; we follow that procedure. The re- spective length of the kinematical curves is recorded in the figure caption. The sensitivity on the chosen kinematical curve is alarmingly large. This observation also implies that the corrections arising from finite geometry can become siz- able in this kinematical configuration.

With respect to the experimental data that this paper at- tempts to describe or to predict, we therefore use the follow- ing theoretical strategy. We employ nonrelativistic kinemat-

ics throughout, i.e., we use the lab cross section as given in Eq. 共13a兲, define the arclength S with nonrelativistic ener- gies, and use the nonrelativistic phase space factor

fps⫽共2␲兲⁴ប² m_N 兩k_␣

i兩

冕

^{d3k␣fk␥2fdk␥f}

冉

^{k␤2dSfdk␤f}

冊

⫻␦

冉

^{2mk␣2fN⫹2mk␤2fN⫹2mk␥2fN⫺ed⫺2mk␣2iN}

冊

⫻␦共k_␣

f⫹k_␤

f⫹k_␥

f⫺k_␣

i兲. 共23兲

FIG. 2. Left side: kinematical curves for the relativistic QFS configuration (42.16°,42.16°,180.0°) with relativistic共dashed curve兲and nonrelativistic共dotted curve兲kinematics and for nonrelativistic QFS configuration (42.32°,42.32°,180.0°) with nonrelativistic kinematics 共solid curve兲at 52 MeV deuteron lab energy. Total arclengths are 9.22, 15.34, and 10.09 MeV, respectively. The dot indicates the position of the exact QFS point. Right side: deuteron tensor analyzing power A_xx(d) as a function of the fractional arclength S/S_max along the kinematical curve for QFS configurations of nucleon-deuteron breakup at 52 MeV deuteron lab energy. As in all calculations of this paper, the results are obtained with a nonrelativistic arclength S. Results for nonrelativistic QFS configuration (42.32°,42.32°,180.0°)共solid curve兲 and for relativistic QFS configuration (42.16°,42.16°,180.0°) 共dotted curve兲are compared.

FIG. 1. Left side: kinematical curves for the relativistic space star configuration (53.5°,53.5°,120.0°) with relativistic共dashed curve兲and nonrelativistic共dotted curve兲kinematics and for the nonrelativistic space star configuration (54.0°,54.0°,120.0°) with nonrelativistic kine- matics共solid curve兲at 65 MeV nucleon lab energy. The total arclengths are 62.92, 63.64, and 63.04 MeV, respectively. The solid and dashed curves are almost indistinguishable in the plot. The dot indicates the position of the exact space star point. Right side: differential cross section as a function of the arclength S along the kinematical curve for the space star configurations of nucleon-deuteron breakup at 65 MeV nucleon lab energy. As in all calculations of this paper, the results are obtained with a nonrelativistic arclength S. Results for the nonrela- tivistic space star configuration (54.0°,54.0°,120.0°) 共solid curve兲 and for the relativistic space star configuration (53.5°,53.5°,120.0°) 共dotted curve兲are compared.

FIG. 3. Differential cross section and nucleon analyzing power A_y(n) as functions of the arclength S along the kinematical curve for various configurations of nucleon-deuteron breakup at 13 MeV nucleon lab energy.共a兲,共b兲 space star configuration (50.5°,50.5°,120.0°), 共c兲,共d兲 collinearity configuration (50.5°,62.5°,180.0°), 共e兲,共f兲 FSI configuration (39.0°,62.5°,180.0°), and 共g兲,共h兲 QFS configuration (39.0°,39.0°,180.0°). Results of the coupled-channel potential with⌬-isobar excitation共solid curve兲are compared with results of the Paris potential 共dashed curve兲. The experimental data are from Ref.关10兴 referring to neutron-deuteron scattering共circles兲 and from Ref. 关11兴 referring to proton-deuteron scattering共crosses兲.

We also note that the fit of the underlying baryonic potentials to data is based on a corresponding entirely nonrelativistic phase space factor. Thus, internal consistency requires the use of the nonrelativistic phase space factor 共23兲. Further- more, that form of description is natural for experimental data that are derived from a nonrelativistic analysis. If, how- ever, the analysis of experimental data is relativistic, we meet the chosen particular kinematic configurations of the experi- ment, such as space star, collinearity, final-state interaction 共FSI兲, or QFS in nonrelativistic kinematics only by an appro- priate change of scattering angles, thereby approximating the relativistic kinematical curves nonrelativistically and scaling the resulting arclengths to the value of the relativistic length.

In case the experimental data do not refer to a particular

kinematic configuration, we still change the scattering angles slightly till the agreement of relativistic and nonrelativistic kinematical curves is significantly improved.

III. RESULTS

Observables of breakup in nucleon-deuteron scattering are calculated for 13 MeV and 65 MeV nucleon lab energy and for 52 MeV deuteron lab energy. The calculations are based on the coupled-channel two-baryon potential A2, defined in Ref. 关2兴; it allows for single ⌬-isobar excitation. Its nucle- onic reference potential, being almost phase equivalent to A2 at low energies, is the Paris potential关7兴. Both potentials are FIG. 4. Deuteron analyzing powers A_y(d), A_{y y}(d), and A_xx(d) as functions of the fractional arclength S/S_maxalong the kinematical curve for various configurations of nucleon-deuteron breakup at 52 MeV deuteron lab energy.共a兲–共c兲configuration (32.5°,32.5°,180.0°) and共d兲–共f兲configuration (37.0°,37.0°,180.0°). Results of the coupled-channel potential with⌬-isobar excitation共solid curve兲are compared with results of the Paris potential共dashed curve兲. Since the experimental analysis uses an arclength S based on relativistic kinematics, the theoretical nonrelativistic description resorts to the identification procedure of Sec. II E. The following scattering angles were used for the calculation 共the ratio of the total relativistic over nonrelativistic arclengths is given simultaneously in square brackets兲: 共a兲–共c兲 (32.7°,32.7°,180.0°)关1.002兴 and共d兲–共f兲(37.2°,37.2°,180.0°)关1.002兴.

used in order to maintain consistency with papers I and II.

Both potentials are taken into account in partial waves up to two-baryon total angular momentum I⫽4. Channel coupling to the ⌬ isobar is considered in all isospin triplet partial waves up to I⫽2. The symmetrized breakup transition ma- trix 具␾0(p_fq_f)␯0(m_f)兩U₀(E_i⫹i0)兩␾␣(q_i)␯␣_i典 to be calcu- lated is expanded into three-body partial waves; the expan- sion is terminated at the three-body total angular momentum J⫽²⁷2 . Any additional three-body partial wave J yields changes not visible in plots.

The calculations are done without Coulomb interaction between protons, they therefore refer to neutron-deuteron

breakup. Nevertheless, results are freely compared to proton- deuteron experiments. Kinematic regions, in which both pro- tons in the final state have small relative momenta and which therefore could see the Coulomb repulsion between the pro- tons, do not occur in the presented plots.

Results for spin-averaged and spin-dependent observables at 13 MeV nucleon lab energy are given in Fig. 3. The ex- perimental data appear analyzed in Refs.关10,11兴nonrelativ- istically. The theoretical predictions of this paper do not need any readjustment of the nonrelativistic kinematical curves for a sound comparison. Anyhow, at this energy the differ- ence between the relativistic and the nonrelativistic kinemati- FIG. 5. Deuteron analyzing powers A_y(d), A_{y y}(d), and A_xx(d) as functions of the fractional arclength S/S_maxalong the kinematical curve for various configurations of nucleon-deuteron breakup at 52 MeV deuteron lab energy.共a兲–共c兲configuration (38.7°,38.7°,180.0°) and共d兲–共f兲configuration (41.0°,41.0°,180.0°). Results of the coupled-channel potential with⌬-isobar excitation共solid curve兲are compared with results of the Paris potential共dashed curve兲. The experimental data are from Ref.关13兴and refer to proton-deuteron scattering; they are given there as functions of the arclength S measured clockwise along the kinematical curve. The A_y(d) data are therefore readjusted to match our convention of a counterclockwise S. Furthermore, since the experimental analysis uses an arclength S based on relativistic kinematics, the theoretical nonrelativistic description resorts to the identification procedure of Sec. II E. The following scattering angles were used for the calculation 共the ratio of the total relativistic over nonrelativistic arclengths is given simultaneously in square brackets兲: 共a兲–共c兲 (38.9°,38.9°,180.0°)关1.004兴 and共d兲–共f兲(41.2°,41.2°,180.0°)关1.002兴.

cal curves is extremely small. The disagreement between the theoretical predictions and the experimental data is most striking for the differential cross section in the space star configuration of Fig. 3共a兲. The experimental data for proton- deuteron and neutron-deuteron breakup are surprisingly far apart. Neither data set is accounted for by theory as has been already observed by others关5兴. Furthermore, the calculations are unable to reproduce the height of the differential cross section peaks at arclength S around 10 MeV in the collinear- ity and in the FSI configurations of Figs. 3共c兲and 3共e兲. This fact is a particular feature of the chosen potentials; additional calculations with more modern potentials are able to remove

that discrepancy关12兴. In the studied observables the effect of the⌬ isobar and of its mediated three-nucleon force is irrel- evant; there is a mild, but nonbeneficial, effect on the central peak of the differential cross section in the QFS configura- tion of Fig. 3共g兲.

Results for deuteron analyzing powers of deuteron-proton scattering at 52 MeV deuteron lab energy are given in Figs. 4 and 5. The experimental data in Figs. 5共a兲–5共c兲are from Ref.

关13兴. There exist also new, but still preliminary, experimental data关14兴for all observables of Figs. 4 and 5. The agreement between our theoretical predictions and these new data ap- pears by and large satisfactory. The new data are not ready- FIG. 6. Differential cross section and nucleon analyzing power A_y(n) as functions of the arclength S along the kinematical curve for various configurations of nucleon-deuteron breakup at 65 MeV nucleon lab energy.共a兲,共b兲 space star configuration (54.0°,54.0°,120.0°), 共c兲,共d兲coplanar star configuration (35.2°,35.2°,180.0°), and共e兲,共f兲QFS configuration (44.0°,44.0°,180.0°). Results of the coupled-channel potential with⌬-isobar excitation共solid curve兲are compared with results of the Paris potential共dashed curve兲. The experimental data are from Refs. 关15,16兴 and refer to proton-deuteron scattering. Since the experimental analysis uses an arclength S based on relativistic kinematics, the theoretical nonrelativistic description has to resort to the identification procedure of Sec. II E. The following scattering angles were used for the calculation 共the ratio of the total relativistic over nonrelativistic arclengths is given simultaneously in square brackets兲: 共a兲,共b兲(54.5°,54.5°,120.0°)关0.999兴, 共c兲,共d兲(35.5°,35.5°,180.0°)关0.999兴, and共e兲,共f兲(44.5°,44.5°,180.0°)关0.997兴.

yet for publication; our predictions are given for further ref- erence. The experimental data of Refs.关13,14兴are and will be analyzed with relativistic kinematics; the identification procedure described in Sec. II E is used. As discussed there, data and results for the configuration (41.0°,41.0°,180.0°) are most affected. On the other hand, the⌬-isobar effects on the considered observables remain small.

Results for spin-averaged and spin-dependent observables at 65 MeV nucleon lab energy are given in Figs. 6 and 7. All experimental data refer to proton-deuteron scattering. The experimental setup realizes the particular scattering configu- rations such as space star, coplanar star, QFS, and collinear- ity within nonrelativistic kinematics. In contrast, the arc-

length S employed for presenting data is derived from rela- tivistic kinematics; thus, the identification procedure for the arclength described in Sec. II E has to be used. The agree- ment between theoretical predictions and experimental data is satisfactory. The effects of the⌬isobar and of its mediated three-nucleon force become more noticeable in some observ- ables, e.g., for the differential cross section in space star and collinear configurations. For some other observables, e.g., for the differential cross section in coplanar star and QFS configurations, the total ⌬-isobar effects are dominated by the dispersive two-body effect; that aspect is worrisome and needs further investigation. The ⌬-isobar effects are not al- ways beneficial.

FIG. 7. Differential cross section and nucleon analyzing power A_y(n) as functions of the arclength S along the kinematical curve for various configurations of nucleon-deuteron breakup at 65 MeV nucleon lab energy.共a兲,共b兲 collinear configuration (30.0°,98.0°,180.0°), 共c兲,共d兲 collinear configuration (59.5°,59.5°,180.0°), and 共e兲,共f兲 nonspecific configuration (20.0°,45.0°,180.0°). Results of the coupled- channel potential with⌬-isobar excitation共solid curve兲 are compared with results of the Paris potential共dashed curve兲. The experimental data are from Refs.关9,17兴and refer to proton-deuteron scattering. Since the experimental analysis uses an arclength S based on relativistic kinematics, the theoretical nonrelativistic description has to resort to the identification procedure of Sec. II E. The following scattering angles were used for the calculation 共the ratio of the total relativistic over nonrelativistic arclengths is given simultaneously in square brackets兲: 共a兲,共b兲(30.3°,98.9°,180.0°)关1.000兴, 共c兲,共d兲(60.0°,60.0°,180.0°)关0.993兴, and共e兲,共f兲(20.2°,45.5°,180.0°)关0.999兴.